Some properties of one-pebble Turing machines with sublogarithmic space

This paper investigates some aspects of the accepting powers of deterministic, nondeterministic, and alternating one-pebble Turing machines with spaces between loglogn and logn. We first investigate a relationship between the accepting powers of two-way deterministic one-counter automata and deterministic (or nondeterministic) one-pebble Turing machines, and show that they are incomparable. Then we investigate a relationship between nondeterminism and alternation, and show that there exists a language accepted by a strongly loglogn space-bounded alternating one-pebble Turing machine, but not accepted by any weakly o(logn) space-bounded nondeterministic one-pebble Turing machine. Finally, we investigate a space hierarchy, and show that for any one-pebble (fully) space constructible function L(n)⩽logn, and for any function L′(n)=o(L(n)), there exists a language accepted by a strongly L(n) space-bounded deterministic one-pebble Turing machine, but not accepted by any weakly L′(n) space-bounded nondeterministic one-pebble Turing machine.